Let \( a = 9 \), \( c = 15 \), solve for \( b \): \( 9^2 + b^2 = 15^2 \).
["Solving for ( b ) in the Equation: ( 9^2 + b^2 = 15^2 ) – A Step-by-Step Algebra Guide", "In algebra, solving equations involving squares is a fundamental skill. One classic problem is finding the value of an unknown ( b ) in the equation:", "[\n9^2 + b^2 = 15^2\n]", "This equation appears frequently in geometry and classic problem-solving contexts. Let’s walk through how to solve it step by step—perfect for students and math enthusiasts looking to strengthen their algebra skills.", "---", "### The Given Values", "We are told:", "- ( a = 9 ), though in this specific equation, only ( 9^2 ) appears, so no dependencies on ( a ) here.\n- ( c = 15 ), equally a constant term.\n- The unknown is ( b ), squared and added to ( 9^2 ), equaling ( 15^2 ).", "### Step 1: Compute the Squares", "Start by calculating the squares:", "[\n9^2 = 81\n]\n[\n15^2 = 225\n]", "Now substitute these into the equation:", "[\n81 + b^2 = 225\n]", "### Step 2: Isolate ( b^2 )", "Subtract 81 from both sides to isolate the term with ( b ):", "[\nb^2 = 225 - 81\n]\n[\nb^2 = 144\n]", "### Step 3: Solve for ( b )", "Now take the square root of both sides:", "[\nb = \pm \sqrt{144}\n]\n[\nb = \pm 12\n]", "### Conclusion", "The solutions are ( b = 12 ) and ( b = -12 ). Both values satisfy the original equation since squaring removes the sign.", "---", "### Why This Equation Matters", "This simple Pythagorean-style equation reflects a right triangle relationship:\n[\n9^2 + b^2 = 15^2\n]\nrepresents the standard form where ( 9 ) and ( b ) are two legs, and ( 15 ) is the hypotenuse. Finding ( b ) completes the triangle, illustrating how algebra bridges numbers and geometry.", "Whether you're solving for side lengths, analyzing quadratic patterns, or preparing for higher math, mastering such equations is essential.", "---", "### Key Takeaways", "- Substitute known squares accurately.\n- Use inverse operations to isolate the variable.\n- Remember square roots yield both positive and negative solutions.\n- Visualize problems geometrically to deepen understanding.", "---", "Try solving similar equations like ( 5^2 + b^2 = 13^2 ) or ( 7^2 + b^2 = 25^2 ) to sharpen your skills. With consistent practice, these step-by-step methods become second nature—empowering you to solve increasingly complex algebra problems with confidence.", "---", "Keywords: solve for b, equation with squares, 9² + b² = 15², Pythagorean theorem problem, algebraic equation solutions, step-by-step algebra, solving quadratic expressions, math practice problems."]
